Positive semidefinite rank
TLDR
The positive semidefinite rank (psd rank) as discussed by the authors is the smallest integer k for which there exist polyhedra of size k = 1 such that the polyhedron is polyhedrically connected with the rank of k. The psd rank has many appealing geometric interpretations, including semidefinite representations of polyhedras and information-theoretic applications.Abstract:
Let $$M \in \mathbb {R}^{p \times q}$$M?Rp×q be a nonnegative matrix. The positive semidefinite rank (psd rank) of M is the smallest integer k for which there exist positive semidefinite matrices $$A_i, B_j$$Ai,Bj of size $$k \times k$$k×k such that $$M_{ij} = {{\mathrm{trace}}}(A_i B_j)$$Mij=trace(AiBj). The psd rank has many appealing geometric interpretations, including semidefinite representations of polyhedra and information-theoretic applications. In this paper we develop and survey the main mathematical properties of psd rank, including its geometry, relationships with other rank notions, and computational and algorithmic aspects.read more
Citations
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Proceedings ArticleDOI
Lower Bounds on the Size of Semidefinite Programming Relaxations
TL;DR: In particular, this paper showed that the cut, TSP, and stable set polytopes on n-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than 2nδ, for some constant δ > 0.
Journal ArticleDOI
Heuristics for exact nonnegative matrix factorization
TL;DR: In this paper, two heuristics for exact nonnegative matrix factorization (exact NMF) are proposed, one inspired from simulated annealing and the other from the greedy randomized adaptive search procedure.
Posted Content
Introduction to Nonnegative Matrix Factorization.
TL;DR: Several aspects ofNMF are discussed, namely, the application in hyperspectral imaging, geometry and uniqueness of NMF solutions, complexity, algorithms, and its link with extended formulations of polyhedra.
Posted Content
A universality theorem for nonnegative matrix factorizations
TL;DR: It is shown that every bounded semialgebraic set $U$ is rationally equivalent to the set of nonnegative size-$k$ factorizations of some matrix $A$ up to a permutation of matrices in the factorization.
Journal ArticleDOI
Classical Information Storage in an n -Level Quantum System
TL;DR: It is shown that whatever the probability distribution of x and the reward function f are, when using a quantum n-level system, the maximum expected reward obtainable with the best possible team strategy is equal to that obtained with the use of a classical n-state system.
References
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